# Direction Cosine Matrix to Quaternions

Convert direction cosine matrix to quaternion vector

## Library

Utilities/Axes Transformations

## Description

The Direction Cosine Matrix to Quaternions block transforms a 3-by-3 direction cosine matrix (DCM) into a four-element unit quaternion vector (q0, q1, q2, q3). The DCM performs the coordinate transformation of a vector in inertial axes to a vector in body axes.

The DCM is defined as a function of a unit quaternion vector by the following:

$DCM=\left[\begin{array}{ccc}\left({q}_{0}^{2}+{q}_{1}^{2}-{q}_{2}^{2}-{q}_{3}^{2}\right)& 2\left({q}_{1}{q}_{2}+{q}_{0}{q}_{3}\right)& 2\left({q}_{1}{q}_{3}-{q}_{0}{q}_{2}\right)\\ 2\left({q}_{1}{q}_{2}-{q}_{0}{q}_{3}\right)& \left({q}_{0}^{2}-{q}_{1}^{2}+{q}_{2}^{2}-{q}_{3}^{2}\right)& 2\left({q}_{2}{q}_{3}+{q}_{0}{q}_{1}\right)\\ 2\left({q}_{1}{q}_{3}+{q}_{0}{q}_{2}\right)& 2\left({q}_{2}{q}_{3}-{q}_{0}{q}_{1}\right)& \left({q}_{0}^{2}-{q}_{1}^{2}-{q}_{2}^{2}+{q}_{3}^{2}\right)\end{array}\right]$

Using this representation of the DCM, there are a number of calculations to arrive at the correct quaternion. The first of these is to calculate the trace of the DCM to determine which algorithms are used. If the trace is greater that zero, the quaternion can be automatically calculated. When the trace is less than or equal to zero, the major diagonal element of the DCM with the greatest value must be identified to determine the final algorithm used to calculate the quaternion. Once the major diagonal element is identified, the quaternion is calculated. For a detailed view of these algorithms, look under the mask of this block.

## Inputs and Outputs

InputDimension TypeDescription

First

3-by-3 direction cosine matrix

OutputDimension TypeDescription

First

4-by-1 quaternion vector